Of course your calves hurt. Keep that up and you can injure yourself, RSI. Counting paces, try walking 100, running 50, etc. Gradually shift that until you're running 100, walking 50, etc. Stretch after your run. Cut back the running until your calf soreness goes away, or almost away. It's not OK. Don't reach forward with your foot. Make contact right under your body. Try for a sense of falling forward and just catching yourself. Plenty of follow-through behind you. Let your foot come up high with a well-bent knee. Let your hips move forward in a straight line, not up and down.
As far as the biking goes, it takes time to get stronger. It's probably not your technique as much as just lack of time in the saddle. Let distance=strength be your motto. When you've worked up to riding 150 miles/week, you'll be plenty strong enough for ego purposes even if the studly still ride past you. 80 mile rides are the ticket to strength. All that interval and strength training stuff is unnecessary. Just ride long hilly routes and it's all taken care of automatically. You want to see lung bits on your shoes once a week.
Power necessary to produce a speed on the bike goes up as the cube of the speed. It's really hard to get a 1 mph faster average. 2 mph faster is a heroic change. An 18 mph cruise on the flat is a good pace. There aren't many people who can cruise solo at 20. 19.3 was all I could ever do, hour after hour. 25 is the realm of the very talented and well trained.
On technique, try to hold your upper body completely still. You shouldn't move back and forth. Your front wheel should also be completely still. Push forward at the top, pull back at the bottom. Try not to have a sense of pushing down. Don't pull up on the backstroke, just unweight the pedal. Let your heel drop naturally, don't tense your calf. Try for a sense of pedaling with your heel cups. One can improve pedal stroke by pedaling very, very fast in a very low gear. If you try that, have a sense of a cushion of air between the bottom of your foot and the insole of your shoe. Try to pedal just one perfect stroke. Then try for 3 in a row, etc.